Voxel-Based Morphometry John Ashburner Wellcome Trust Centre for Neuroimaging, 12 Queen Square, London, UK. Inter-subject variability ...diverse and dissimilar fish [brains] can be referred as a whole to identical functions of very different coordinate systems... D’Arcy Thompson (1917). GROWTH AND FORM Morphometry is defined as: Measurement of the form of organisms or of their parts. The American Heritage® Medical Dictionary What kind of differences are we looking for? * Usually, we try to localise regions of difference. * Mass-univariate models. * Using methods similar to SPM * Typically localising volumetric differences * Some anatomical differences can not be localised. * Need multivariate models. * Differences in terms of proportions among measurements. * This talk is about a mass-univariate method - VBM. Voxel-Based Morphometry - VBM * Based on comparing regional volumes of tissue. * Suitable for studying focal volumetric differences of grey matter. * Assumes independence among voxels * Shows differences that are easier to interpret * Produce a map of statistically significant differences among populations of subjects. * e.g. compare a patient group with a control group. * or identify correlations with age, test-score etc. * The data are pre-processed to sensitise the tests to regional tissue volumes. Group-wise statistics SPM for group fMRI fMRI time-series Preprocessing Stat. modelling Results query “Contrast” spm T Image Preprocessing Stat. modelling Results query “Contrast” Image Preprocessing Stat. modelling Results query “Contrast” Image fMRI time-series fMRI time-series SPM for structural MRI High-res T1 MRI ? High-res T1 MRI ? High-res T1 MRI ? ? Group-wise statistics The need for tissue segmentation * High-resolution MRI reveals fine structural detail in the brain, but not all of it reliable or interesting * Noise, intensity-inhomogeneity, vasculature, … * MR Intensity is usually not quantitatively meaningful (in the same way that e.g. CT is) * fMRI time-series allow signal changes to be analysed statistically, compared to baseline or global values * Regional volumes of the three main tissue types: gray matter, white matter and CSF, are well-defined and potentially very interesting Volumetry T1-Weighted MRI Grey Matter Spatial Normalisation * Brains of different subjects vary in shape and size. * Need to bring them all into a common anatomical space. * Examine homologous regions across subjects * Improve anatomical specificity * Improve sensitivity * Report findings in a common anatomical space (eg MNI space) * In SPM, alignment is achieved by matching grey matter with grey matter and white matter with white matter. * Need to segment. Original Warped Template “Modulation” – change of variables. Deformation Field Jacobians determinants Encode relative volumes. Smoothing Each voxel after smoothing effectively becomes the result of applying a weighted region of interest (ROI). Before convolution Convolved with a circle Convolved with a Gaussian Statistical Parametric Mapping… – group 1 voxel by voxel modelling parameter estimate standard error = statistic image or SPM group 2 Some Explanations of the Differences Mis-classify Mis-register Folding Thickening Thinning Mis-register Mis-classify VBM Pre-processing in SPM8 * Use New Segment for characterising intensity distributions of tissue classes, and writing out “imported” images that DARTEL can use. * Run DARTEL to estimate all the deformations. * DARTEL warping to generate smoothed, “modulated”, warped grey matter. * Statistics. New Segment • Generate low • resolution GM and WM images for each subject (“DARTEL imported”). Generate full resolution GM map for each subject. Segmentation * Segmentation in SPM8 also estimates a spatial transformation that can be used for spatially normalising images. * It uses a generative model, which involves: * Mixture of Gaussians (MOG) * Bias Correction Component * Warping (Non-linear Registration) Component Image Intensity Distributions (T1-weighted MRI) Mixture of Gaussians (MOG) * Classification is based on a Mixture of Gaussians model (MOG), which represents the intensity probability density by a number of Gaussian distributions. Frequency Image Intensity Belonging Probabilities Belonging probabilities are assigned by normalising to one. Non-Gaussian Intensity Distributions * Multiple Gaussians per tissue class allow non-Gaussian intensity distributions to be modelled. * E.g. accounting for partial volume effects Modelling a Bias Field * A bias field is modelled as a linear combination of basis functions. Corrupted image Bias Field Corrected image New Segmentation – TPMs Segment button New Seg Toolbox New Segmentation – registration Segment button * 9×10×9 × 3 = 2,430 New Seg Toolbox * 59×70×59 × 3 = 731,010 Deforming the Tissue Probability Maps * Tissue probability images are deformed so that they can be overlaid on top of the image to segment. Optimisation * The “best” parameters are those that minimise this objective function. * Optimisation involves finding them. * Begin with starting estimates, and repeatedly change them so that the objective function decreases each time. Searching for the optimum Start Optimum Alternate between optimising different groups of parameters Limitations of the current model * Assumes that the brain consists of only the tissues modelled by the TPMs * No allowance for lesions (stroke, tumours, etc) * Prior probability model is based on relatively young and healthy brains * Less appropriate for subjects outside this population * Needs reasonable quality images to work with * No severe artefacts * Good separation of intensities * Good initial alignment with TPMs... Run DARTEL (create Templates) • Simultaneously • align “DARTEL imported” GM and WM for all subjects. Generates templates and parameterisations of relative shapes. DARTEL Image Registration * Uses fast approximations * Deformation integrated using scaling and squaring * Uses Levenberg-Marquardt optimiser Grey matter template warped to individual * Multi-grid matrix solver * Matches GM with GM, WM with WM etc * Diffeomorphic registration takes about 30 mins per image pair (121×145×121 images). Individual scan Evaluations of nonlinear registration algorithms One-to-one mapping One-to-one mappings (no folding of the deformations). Positive Jacobian determinants (relative volumes). Hard to achieve with the usual small deformation models. Jacobian determinants (should be positive) Displacements don’t add linearly Forward Inverse Composed Subtracted DARTEL * Parameterising the deformation * φ(0) = Identity 1 * φ(1) = ∫ u(φ(t))dt * u is a flow field to be estimated t=0 * Scaling and squaring is used to generate deformations. Scaling and squaring example Registration objective function * Simultaneously minimize the sum of: * Matching Term * * * Regularisation term * * * Drives the matching of the images. Multinomial assumption A measure of deformation roughness Regularises the registration. A balance between the two terms. Effect of Different Regularisation Terms Simultaneous registration of GM to GM and WM to WM Subject 1 Grey matter White matter Grey matter White matter Grey matter White matter Grey matter Template Grey matter White matter White matter Subject 2 Subject 4 Subject 3 Template Initial Average Iteratively generated from all subjects in study Begin with rigidly aligned tissue probability maps After a few iterations Final template Initial GM images Warped GM images Normalise to MNI Space • Use shape parameterisations to generate smoothed Jacobian scaled and spatially normalised GM images for each subject. MNI Space Grey matter average of 452 subjects – affine Population Space Grey matter average of 471 subjects - nonlin “Modulation” – change of variables. Deformation Field Jacobians determinants Encode relative volumes. Smoothing * The analysis will be most sensitive to effects that match the shape and size of the kernel * The data will be more Gaussian and closer to a continuous random field for larger kernels * Results will be rough and noise-like if too little smoothing is used * Too much will lead to distributed, indistinct blobs Smoothing * Between 7 and 14mm is probably best * (lower is okay with better registration, e.g. DARTEL) * The results below show two fairly extreme choices, 5mm on the left, and 16mm, right Some References * Ashburner & Friston. “Voxel-based morphometry-the methods”. Neuroimage 11(6):805-821, 2000. * Ashburner & Friston. “Unified Segmentation”. NeuroImage 26:839-851, 2005. * Ashburner. “A Fast Diffeomorphic Image Registration Algorithm”. NeuroImage 38:95-113 (2007). * Ashburner & Friston. “Computing Average Shaped Tissue Probability Templates”. NeuroImage 45:333-341, 2009. * Klein et al. Evaluation of 14 nonlinear deformation algorithms applied to human brain MRI registration. NeuroImage 46(3):786-802 (2009). * Ashburner. “Computational Anatomy with the SPM software”. Magnetic Resonance Imaging 27(8):1163-1174, 2009. A brief history of VBM with SPM * A Voxel-Based Method for the Statistical Analysis of Gray and White Matter Density… Wright, McGuire, Poline, Travere, Murrary, Frith, Frackowiak & Friston. NeuroImage 2(4), 1995 (!) * * Voxel-Based Morphometry – The Methods. Ashburner and Friston. NeuroImage 11(6), 2000 * * * “Optimised” GM-normalisation (ad hoc procedure), modulation of segments with Jacobian determinants Unified Segmentation. Ashburner and Friston. NeuroImage 26(3), 2005 * * Non-linear spatial normalisation, automatic segmentation Thorough consideration of assumptions and confounds A Voxel-Based Morphometric Study of Ageing… Good, Johnsrude, Ashburner, Henson and Friston. NeuroImage 14(1), 2001 * * Rigid reorientation (by eye), semi-automatic scalp editing and segmentation, 8mm smoothing, SPM statistics, global covars. Principled generative model for segmentation using deformable priors A Fast Diffeomorphic Image Registration Algorithm. Ashburner. Neuroimage 38(1), 2007 * Large deformation normalisation to average shape templates General Linear Model a1 ( x, y, z ) a ( x, y , z ) 2 Y X e xyz xyz a N ( x, y , z ) 1 1 X 0 0 0 0 2 exyz ~ N (0, xyzV ) 1 1 “Globals” for VBM * Shape is really a multivariate concept * Dependencies among volumes in different regions * SPM is mass univariate * Combining voxel-wise information with “global” integrated tissue volume provides a compromise * Using either ANCOVA or proportional scaling Figures from: Voxel-based morphometry of the human brain… Mechelli, Price, Friston and Ashburner. Current Medical Imaging Reviews 1(2), 2005. Above: (ii) is globally thicker, but locally thinner than (i) – either of these effects may be of interest to us. Below: The two “cortices” on the right both have equal volume… Total Intracranial Volume (TIV/ICV) * “Global” integrated tissue volume may be correlated with interesting regional effects * Correcting for globals in this case may overly reduce sensitivity to local differences * Total intracranial volume integrates GM, WM and CSF, or attempts to measure the skull-volume directly * Not sensitive to global reduction of GM+WM (cancelled out by CSF expansion – skull is fixed!) * Correcting for TIV in VBM statistics may give more powerful and/or more interpretable results * See also Pell et al (2009) doi:10.1016/j.neuroimage.2008.02.050 VBM’s statistical validity * Residuals are not normally distributed * Little impact on uncorrected statistics for experiments comparing reasonably sized groups * Probably invalid for experiments that compare single subjects or tiny groups with a larger control group * Need to use nonparametric tests that make less assumptions, e.g. permutation testing with SnPM VBM’s statistical validity * Correction for multiple comparisons * RFT correction based on peak heights should be OK * Correction using cluster extents is problematic * SPM usually assumes that the smoothness of the residuals is spatially stationary * VBM residuals have spatially varying smoothness * Bigger blobs expected in smoother regions * Toolboxes are now available for non-stationary cluster-based correction * http://www.fmri.wfubmc.edu/cms/NS-General Longitudinal VBM * The simplest method for longitudinal VBM is to use cross-sectional preprocessing, but longitudinal statistical analyses * Standard preprocessing not optimal, but unbiased * Non-longitudinal statistics would be severely biased * (Estimates of standard errors would be too small) * Simplest longitudinal statistical analysis: two-stage summary statistic approach (common in fMRI) * Within subject longitudinal differences or beta estimates from linear regressions against time Longitudinal VBM variations * Intra-subject registration over time is much more accurate than inter-subject normalisation * Different approaches suggested to capitalise * A simple approach is to apply one set of normalisation parameters (e.g. Estimated from baseline images) to both baseline and repeat(s) * Draganski et al (2004) Nature 427: 311-312 * “Voxel Compression mapping” – separates expansion and contraction before smoothing * Scahill et al (2002) PNAS 99:4703-4707 Longitudinal VBM variations * Can also multiply longitudinal volume change with baseline or average grey matter density * Chételat et al (2005) NeuroImage 27:934-946 * Kipps et al (2005) JNNP 76:650 * Hobbs et al (2009) doi:10.1136/jnnp.2009.190702 * Note that use of baseline (or repeat) instead of average might lead to bias * Thomas et al (2009) doi:10.1016/j.neuroimage.2009.05.097 * Unfortunately, the explanations in this reference relating to interpolation differences are not quite right... there are several open questions here... Alzheimer’s Disease example Baseline Image Standard clinical MRI 1.5T T1 SPGR 1x1x1.5mm voxels Repeat image 12 month follow-up rigidly registered Subtraction image Longitudinal VBM variations Late Early Late CSF - Early CSF Late CSF Early CSF Late CSF - modulated CSF Smoothed Warped early Difference Relative volumes CSF “modulated” by relative volume

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